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This is probably not news to folks on the right [[not necessarily correct]] side of the ditch, but it is news to me.

<2015 “Whether because of the resemblance to Mathew, or because he had taken Strike’s chair, or because Strike genuinely senses a tosser when he saw one, Strike found the youth obscurely objectionable.”—Career of Evil by Robert Galbraith (aka J. K. Rowling), page 58>

TOSSERnoun
1) (Britain, slang) A male who masturbates [[excessively?]].
2 (Britain, slang) An objectionable male, often somewhat obnoxious in demeanor.
<That bloke who ripped me off was a right tosser.>
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OXFORD DICTIONARIES

TOSSERnoun: British vulgar slang A person who masturbates (used as a general term of abuse).
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OXFORD ENGLISH DICTIONARY

TOSSERnoun: A term of contempt or abuse for a person; a ‘jerk’. [probably from the senses to masturbate]

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I rarely refer to the Urban Dictionary with its heavy weight of unreviewed dross, but I think they offer a good feel for this one, which has proved hard (at least for me) to put my finger on with its multitude of interpretations:

URBAN DICTIONARY

TOSSER: Literally, one who masturbates [[Seems to me that masturbation has got a bum rap, but I guess it ain,t what it used to be with the threat of blindness, pimples, mental illness, . . .]] Common usage typically refers to anyone of whom you have a low opinion.

Some Words and Descriptions Related to Tosser: wanker, idiot, twat, cunt, loser, dickhead, prick, asshole, jerk, fool, moron, dick, bastard, bullshitter; the word tosser describes a person with a tendency to show off or brag in an excessive and embarrassing way; someone who is loud and rude.

<Well at first I thought this guy delivers, but in the end he was just another tosser.>
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The first quote is from the Oxford English Dictionary with the rest being from newspaper archives:

<1977 “She came on in a big mac and flashed her legs like an old tosser before throwing it off.”—Zigzag, April, page 43>

The Danish noun 'tosse' means 'crazy person', 'someone who isn't in his right mind'. I surmise, evidencelessly, that this term may have entered English via the speech of Danish seafarers and then been transmogrified into the term that Brits all know and love today.

There is another slang term in British English which to me has the feel of being fairly close to 'tosser' (in the sense of 'idiot') in both form and meaning, namely 'tosspot'.

However, Michael Quinion offers a rather different take on that word, which he says meant "a habitual drinker, one who tossed back the contents of his pot to make ready for the next". But regarding 'tosser', Quinion is mute.

Okay, seriously, Ken. "Tosser" pretty well covers the same ground as "wanker" and can essentially be used for anyone whom you dismiss as obnoxious, or useless. If anything, I have the sense that it is even more aggressive and dismissive than "wanker". Among good friends, you could probably get away with "you daft wanker" as an affectionate aside. It would be harder to do that with "you daft tosseer".

Be that as it may, it is more than anything dismissive in tone. Not so much a numbskull or an idiot, but someone who makes my life harder or unpleasant.

As far as the etymology is concerned, I have no idea what "came" first, "tosser" or "toss off", but they are derivationally related.

Quinion is right with "toss-pot", which, as far as I remember, crops up as early as the 16th century. It is possible that "tosser" originally derived from this and subsequently picked up its additional sexual meanings.

Ed: Yes, I was right about "toss-pot":

But when I came unto my beds,
With hey, ho, the wind and the rain,
With toss-pots still had drunken heads,
For the rain it raineth every day.

A great while ago the world begun,
With hey, ho, the wind and the rain,
But that's all one, our play is done,
And we'll strive to please you every day.

An interesting advertising campaign was run here by the Environmental Protection Authority. It played on the slang meaning and the regular meaning of the word tosser. The campaign centred around the phrase Don't be a tosser! and referred to people tossing rubbish out of their car, tossing rubbish in the streets or simply leaving rubbish behind when they left an area. The idea was simply that if you tossed rubbish anywhere but in the bin you were a tosser.